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Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is $$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$ if $X$ and $Y$ take values in the a me …

Let $(X,\mu,\mathcal{F})$ be a probability space. The paper Equiconvergence of Martingales by Edward Boylan introduced a pseudometric on sub-$\sigma$-fields (sub-$\sigma$-algebras) of $\mathcal{F}$ a …

I'm interested in a new (to me) mode of convergence which is stronger than convergence in measure/probability. I want to know if it has a name and if it is used much in the literature. I will write …

I am reasking a year-old math.stackexchange.com question asked by someone else. (For my needs every space $X$ and $Y$ will be Polish---that is a completely separably metrizable space.) The Tietze ex …

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Update (2015-09-28): The property must also be closed under isometric embeddings.) (Sup …

I'm have difficultly nailing down the direction of some implications. For $2 \leq q < \infty$, there are (at least) two ways to say that a Banach space $B$ has "cotype $q$". $B$ has cotype q. $B$ is …

Recall a Polish space is a completely metrizable separable space. Say a Polish space $Y$ is a terminal space if for any Polish space $X$ and any closed $C \subseteq X$, one can extend a continuous m …

Short question If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked …

The Sz.-Nagy dilation theorem says that for a Hilbert space $H$ with nonexpansive operator $T$, there is a larger space $H'$ containing $H$ and a unitary operator $U$ on $H'$ such that for all $x \in …

I'll attempt to answer this with what I've found: (1) The answer to my main question is that it is not true. In this document of Pisier's, he states It is possible to find a uniformly convex s …

Suppose $Y$ is a Banach space and $X$ is a finite-dimensional subspace of $Y$. Further assume $T:X \rightarrow X$ is a linear operator which is power bounded from above and below, in other words ther …